Methods and systems for creating an interest rate swap volatility index and trading derivative products based thereon

ABSTRACT

Systems and methods for creating and disseminating an interest rate swap volatility index based on an underlying interest rate swaption, and for creating and trading derivative investment products based on the interest rate swap volatility index, are disclosed. In one aspect, an interest rate swap volatility index based on an underlying interest rate swaption is calculated. The interest rate swap volatility index may be accessed by a processor of a trading platform and a standardized, exchange traded derivative may be created based on the calculated interest rate swap volatility index. Information associated with the interest rate swap volatility index derivative may then be transmitted for display.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Application Ser. No.61/499,077, filed Jun. 20, 2011, and claims the benefit of U.S.Application Ser. No. 61/577,270, filed Dec. 19, 2011, and the entiretyof each of these aforementioned applications is hereby incorporatedherein by reference.

FIELD OF THE DISCLOSURE

The present disclosure relates to derivative investment markets. Morespecifically, the present disclosure relates to electronically creatingand disseminating one or more volatility indices calculated usinginterest rate swaption (i.e., is an option granting its owner the rightbut not the obligation to enter into an underlying interest rate swap)data, and facilitating the electronic creation and trading of derivativeproducts based on one or more indices relating to volatility.Additionally, the present disclosure relates to electronically creatingand disseminating one or more indices relating to interest rate swapvolatility, and facilitating the electronic creation and trading ofderivative products based on the one or more indices relating tointerest rate swap volatility.

BACKGROUND

A derivative is a financial instrument whose value depends at least inpart on the value and/or characteristic(s) of another security, known asan underlying asset. Examples of underlying assets include, but are notlimited to: interest rate financial instruments (e.g., bonds, interestrate swaps, and interest rate swaptions), commodities, securities,electronically traded funds, and indices. Two exemplary and well knownderivatives are options and futures contracts.

Derivatives, such as options and futures contracts, may be tradedover-the-counter and/or on other trading platforms, such as organizedexchanges (e.g., the Chicago Board Options Exchange, Incorporated(“CBOE”)). In over-the-counter transactions the individual parties to atransaction are able to customize each transaction to meet each party'sindividual needs. With trading platform or exchange traded derivatives,buy and sell orders for standardized derivative contracts are submittedto an exchange where they are matched and executed. Generally, moderntrading exchanges have exchange specific computer systems that allow forthe electronic submission of orders via electronic communicationnetworks, such as the Internet. An example of an exchange specificcomputer system is illustrated in FIG. 1.

Once matched and executed, the executed trade is transmitted to aclearing corporation that stands between the holders and writers ofderivative contracts. When exchange traded derivatives are exercised,the cash or underlying assets are delivered, when necessary, to theclearing corporation and the clearing corporation disperses the assetsas appropriate and defined by the consequence(s) of the trades.

An option contract gives the contract holder a right, but not anobligation, to buy or sell an underlying asset at a specific price on orbefore a certain date, depending on the option style (e.g., American orEuropean). Conversely, an option contract obligates the seller of thecontract to deliver an underlying asset at a specific price on or beforea certain date, depending on the option style (e.g., American orEuropean). An American style option may be exercised at any time priorto its expiration. A European style option may be exercised only at itsexpiration, i.e., at a single pre-defined point in time.

There are generally two types of options: calls and puts. A call optionconveys to the holder a right to purchase an underlying asset at aspecific price (i.e., the strike price), and obligates the writer todeliver the underlying asset to the holder at the strike price. A putoption conveys to the holder a right to sell an underlying asset at aspecific price (i.e., the strike price), and obligates the writer topurchase the underlying asset at the strike price.

There are generally two types of settlement processes: physicalsettlement and cash settlement. During physical settlement, funds aretransferred from one party to another in exchange for the delivery ofthe underlying asset. During cash settlement, funds are delivered fromone party to another according to a calculation that incorporates dataconcerning the underlying asset.

A futures contract gives a buyer of the future an obligation to receivedelivery of an underlying commodity or asset on a fixed date in thefuture. Accordingly, a seller of the future contract has the obligationto deliver the commodity or asset on the specified date for a givenprice. Futures may be settled using physical or cash settlement. Bothoptions and futures contracts may be based on abstract marketindicators, such as indices.

An index is a statistical composite that is used to indicate theperformance of a market or a market sector over various time periods,i.e., act as a performance benchmark. Examples of indices include theDow Jones Industrial Average, the National Association of SecuritiesDealers Automated Quotations (“NASDAQ”) Composite Index, and theStandard & Poor's 500 (“S&P 500”). As noted above, options on indicesare generally cash settled. For example, using cash settlement, a holderof an index call option receives the right to purchase not the indexitself, but rather a cash amount equal to the value of the indexmultiplied by a multiplier, e.g., $100. Thus, if a holder of an indexcall option exercises the option, the writer of the option must pay theholder, provided the option is in-the-money, the difference between thecurrent value of the underlying index and the strike price multiplied bya multiplier.

Among the indices that derivatives may be based on are those that gaugethe volatility of a market or a market subsection. For example, CBOEcreated and disseminates the CBOE Market Volatility Index or VIX®, whichis a key measure of market expectations of near-term volatility conveyedby S&P 500 stock index options prices. Additionally, CBOE offersexchange traded derivative products (both futures and options) that usethe VIX as the underlying asset. Volatility indices and the derivativeproducts based thereon have been widely accepted by the financialindustry as both a useful tool to hedge positions and as a device forexpressing investment views on the direction of volatility.

While several volatility indices exist, there currently exists noimplementation of a volatility gauge for interest rate swap markets thatis theoretically consistent with prices prevailing in existing swaptionmarkets. Particularly, no standardized benchmarks exist to estimate thevolatility in the interest rate swap (“IRS”) markets over a giveninvestment horizon. Because no standardized benchmark currently existsthat reflects expected IRS market volatility, traders, other marketparticipants, and/or money managers currently trade interest rateswaptions (i.e., options on interest rate swaps) to hedge otherfinancial positions, facilitate market-making, and/or take particularinvestment positions related to market volatility. However, thestrategies employed in attempting to hedge risk via the trading ofinterest rate swaptions do not necessarily lead to accurate profits andlosses due to price dependency, i.e., the tendency to generate profitsand losses that are affected by the path of price movements betweentrade inception and expiry dates rather than the absolute price levelprevailing at the time of swaption expiry.

BRIEF SUMMARY

In order to provide an effective volatility index related to theinterest rate swap market, and to provide exchange traded derivativeproducts based on such an index, methods and systems for creating anddisseminating a volatility index based on swaption information isdisclosed. Particularly, the present disclosure sets forth systems andmethods for creating an interest rate swap volatility index and tradingderivative products based thereon.

According to one aspect, a computer-implemented method of calculating aninterest rate swap volatility index is provided. The method includes,using a processor in a trading platform, calculating an interest rateswap volatility index associated with an underlying interest rateswaption. The processor then displays the interest rate swap volatilityindex associated with the underlying interest rate swaption on a tradingplatform display device coupled with the trading platform. Calculatingthe interest rate swap volatility index may include the processoraggregating prices of both at-the-money and out-of-the-money receiverand payer interest rate swaptions, for example in a single equation thatis independent of any option pricing model. The method may reduce arelationship between an implied volatility of the underlying interestrate swaption and a strike of the underlying interest rate swaption intoa single point for each maturity-tenor combination of the underlyinginterest rate swaption.

According to another aspect, a trading platform is disclosed. Thetrading platform may include a display device and a memory storing a setof instructions for calculating an interest rate swap volatility indexassociated with an underlying interest rate swaption. The tradingplatform may further include a processor in communication with thedisplay device and the memory, where the processor is configured toexecute the set of instructions stored in the memory to calculate theinterest rate swap volatility index associated with the underlyinginterest rate swaption, and to display the interest rate swap volatilityindex associated with the underlying interest rate swaption on thedisplay device.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram of a financial exchange's computerized tradingsystem;

FIG. 2 is a diagram of a financial exchange's back end trading system;

FIG. 3 is a flow diagram of a method of calculating a Basis Point IRSvolatility index;

FIG. 4 is a flow diagram of a method of calculating a Percentage IRSvolatility index; and

FIG. 5 is a diagram of a general purpose computer system that can bemodified via computer hardware or software to be customized andspecialized so as to be suitable for use in a financial exchangescomputerized trading system.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The embodiments of the present invention can be implemented on existingfinancial exchange systems and/or other known financial industrysystems. Both financial exchange systems and other known financialindustry systems utilize a combination of computer hardware (e.g.,client and server computers, which may include computer processors,memory, storage, input and output devices, and other known components ofcomputer systems; electronic communication equipment, such as electroniccommunication lines, routers, switches, etc.; electronic informationstorage systems, such as network-attached storage and storage areanetworks) and computer software (i.e., the instructions that cause thecomputer hardware to function in a specific way) to achieve the desiredsystem performance. It should be noted that financial exchange systemsmay be floor-based open outcry systems, pure electronic systems, or somecombination of floor-based open outcry and pure electronic systems.

FIG. 1 illustrates an electronic trading system 100 which may be usedfor creating and disseminating a swaption index (such as an interestrate swap volatility index) and/or creating, listing and tradingderivative contracts that are based on a swaption index. One havingordinary skill in the art would readily understand that system 100, asdescribed in detail below, would be implemented utilizing a combinationof computer hardware and software, as described in the paragraph above.It will be appreciated that the described systems may implement themethods described below.

The system 100 includes components operated by an exchange, as well ascomponents operated by others who access the exchange to execute trades.The components shown within the dashed lines are those operated by theexchange. Components outside the dashed lines are operated by others,but nonetheless are necessary for the operation of a functioningexchange. The exchange components 122 of the trading system 100 includean electronic trading platform 120, a member interface 108, a matchingengine 110, and backend systems 112. Backend systems not operated by theexchange but which are integral to processing trades and settlingcontracts are the Clearing Corporation's systems 114, and Member Firms'backend systems 116.

Market Makers may access the trading platform 120 directly throughpersonal input devices 104 which communicate with the member interface108. Market makers may quote prices for the derivative contracts of thepresent invention, e.g., interest rate swap volatility index derivativecontracts. Non-member Customers 102, however, must access the exchangethrough a Member Firm. Customer orders are routed through Member Firmrouting systems 106. The Member Firm routing systems 106 forward theorders to the exchange via the member interface 108. The memberinterface 108 manages all communications between the Member Firm routingsystems 106 and Market Makers' personal input devices 104; determineswhether orders may be processed by the trading platform; and determinesthe appropriate matching engine for processing the orders. Although onlya single matching engine 110 is shown in system 100, the tradingplatform 120 may include multiple matching engines. Different exchangetraded products may be allocated to different matching engines forefficient execution of trades. When the member interface 102 receives anorder from a Member Firm routing system 106, the member interface 108determines the proper matching engine 110 for processing the order andforwards the order to the appropriate matching engine. The matchingengine 110 executes trades by pairing corresponding marketable buy/sellorders. Non-marketable orders are placed in an electronic order book.

Once orders are executed, the matching engine 110 sends details of theexecuted transactions to the exchange backend systems 112, to theClearing Corporation systems 114, and to the Member Firm backend systems116. The matching engine also updates the order book to reflect changesin the market based on the executed transactions. Orders that previouslywere not marketable may become marketable due to changes in the market.If so, the matching engine 110 executes these orders as well.

The exchange backend systems 112 perform a number of differentfunctions. For example, contract definition and listing data originatewith the Exchange backend systems 112. The swaption indices of thepresent invention, e.g., the interest rate swap volatility indicesdescribed below, and pricing information for derivative contractsassociated with the indices of the present invention are disseminatedfrom the exchange backend systems to market data vendors 118. Customers102, market makers 104, and others may access the market data regardingthe indices of the present invention and the derivative contracts basedon the indices of the present invention via, for example, proprietarynetworks, on-line services, and the like.

The exchange backend systems also evaluate the underlying asset orassets on which the derivative contracts of the present invention arebased. At expiration, the backend systems 112 determine the appropriatesettlement amounts and supply final settlement data to the ClearingCorporation 114. The Clearing Corporation 114 acts as the exchange'sbank and performs a final mark-to-market on Member Firm margin accountsbased on the positions taken by the Member Firms' customers. The finalmark-to-market reflects the final settlement amounts for the derivativecontracts of the present invention, and the Clearing Corporationdebits/credits Member Firms' accounts accordingly. These data are alsoforwarded to the Member Firms' systems 116 so that they may update theircustomer accounts as well.

FIG. 2 shows an embodiment of the exchange backend systems 112 used forcreating and disseminating an index of the present invention, e.g., aninterest rate swap volatility index, and/or creating, listing, andtrading derivative contracts that are based on an index of the presentinvention. A derivative contract of the present invention has adefinition stored in module 202 that contains all relevant dataconcerning the derivative contract to be traded on the trading platform120, including, for example, the contract symbol, a definition of theunderlying asset or assets associated with the derivative, or a term ofa calculation period associated with the derivative. A pricing dataaccumulation and dissemination module 204 receives contract informationfrom the derivative contract definition module 202 and transaction datafrom the matching engine 110. The pricing data accumulation anddissemination module 204 provides the market data regarding open bidsand offers and recent transactions to the market data vendors 118. Thepricing data accumulation and dissemination module 204 also forwardstransaction data to the Clearing Corporation 114 so that the ClearingCorporation 114 may mark-to-market the accounts of Member Firms at theclose of each trading day, taking into account current market prices forthe derivative contracts of the present invention. Finally, a settlementcalculation module 206 receives input from the derivative monitoringmodule 208. On the settlement date the settlement calculation module 206calculates the settlement amount based on the value associated with theunderlying asset or assets, e.g., the value of an interest rate swapvolatility index. The settlement calculation module 206 forwards thesettlement amount to the Clearing Corporation 114, which performs afinal mark-to-market on the Member Firms' accounts to settle thederivative contract of the present invention.

Referring to FIG. 5, an illustrative embodiment of a general computersystem that may be used for one or more of the components shown in FIG.1, or in any other trading system configured to carry out the methodsdiscussed in further detail below, is shown and is designated 500. Thecomputer system 500 can include a set of instructions that can beexecuted to cause the computer system 500 to perform any one or more ofthe methods or computer based functions disclosed herein. The computersystem 500 may operate as a standalone device or may be connected, e.g.,using a network, to other computer systems or peripheral devices.

In a networked deployment, the computer system may operate in thecapacity of a server or as a client user computer in a server-clientuser network environment, or as a peer computer system in a peer-to-peer(or distributed) network environment. The computer system 500 can alsobe implemented as or incorporated into various devices, such as apersonal computer (“PC”), a tablet PC, a set-top box (“STB”), a personaldigital assistant (“PDA”), a mobile device, a palmtop computer, a laptopcomputer, a desktop computer, a network router, switch or bridge, or anyother machine capable of executing a set of instructions (sequential orotherwise) that specify actions to be taken by that machine. In aparticular embodiment, the computer system 500 can be implemented usingelectronic devices that provide voice, video or data communication.Further, while a single computer system 500 is illustrated, the term“system” shall also be taken to include any collection of systems orsub-systems that individually or jointly execute a set, or multiplesets, of instructions to perform one or more computer functions.

As illustrated in FIG. 5, the computer system 500 may include aprocessor 502, such as a central processing unit (“CPU”), a graphicsprocessing unit (“GPU”), or both. Moreover, the computer system 500 caninclude a main memory 504 and a static memory 506 that can communicatewith each other via a bus 508. As shown, the computer system 500 mayfurther include a video display unit 510, such as a liquid crystaldisplay (“LCD”), an organic light emitting diode (“OLED”), a flat paneldisplay, a solid state display, or a cathode ray tube (“CRT”).Additionally, the computer system 500 may include an input device 512,such as a keyboard, and a cursor control device 514, such as a mouse.The computer system 500 can also include a disk drive unit 516, a signalgeneration device 518, such as a speaker or remote control, and anetwork interface device 520.

In a particular embodiment, as depicted in FIG. 5, the disk drive unit516 may include a computer-readable medium 522 in which one or more setsof instructions 524, e.g., software, can be embedded. Further, theinstructions 524 may embody one or more of the methods or logic asdescribed herein. In a particular embodiment, the instructions 524 mayreside completely, or at least partially, within the main memory 504,the static memory 506, and/or within the processor 502 during executionby the computer system 500. The main memory 504 and the processor 502also may include computer-readable media.

In an alternative embodiment, dedicated hardware implementations, suchas application specific integrated circuits, programmable logic arraysand other hardware devices, can be constructed to implement one or moreof the methods described herein. Applications that may include theapparatus and systems of various embodiments can broadly include avariety of electronic and computer systems. One or more embodimentsdescribed herein may implement functions using two or more specificinterconnected hardware modules or devices with related control and datasignals that can be communicated between and through the modules, or asportions of an application-specific integrated circuit. Accordingly, thepresent system encompasses software, firmware, and hardwareimplementations.

In accordance with various embodiments of the present disclosure, themethods described herein may be implemented by software programsexecutable by a computer system. Further, in an exemplary, non-limitedembodiment, implementations can include distributed processing,component/object distributed processing, and parallel processing.Alternatively, virtual computer system processing can be constructed toimplement one or more of the methods or functionality as describedherein.

The present disclosure contemplates a computer-readable medium thatincludes instructions 524 or receives and executes instructions 524responsive to a propagated signal, so that a device connected to anetwork 526 can communicate voice, video or data over the network 526.Further, the instructions 524 may be transmitted or received over thenetwork 526 via the network interface device 520.

While the computer-readable medium is shown to be a single medium, theterm “computer-readable medium” includes a single medium or multiplemedia, such as a centralized or distributed database, and/or associatedcaches and servers that store one or more sets of instructions. The term“computer-readable medium” shall also include any medium that is capableof storing, encoding or carrying a set of instructions for execution bya processor or that cause a computer system to perform any one or moreof the methods or operations disclosed herein.

In a particular non-limiting, exemplary embodiment, thecomputer-readable medium can include a solid-state memory such as amemory card or other package that houses one or more non-volatileread-only memories. Further, the computer-readable medium can be arandom access memory or other volatile re-writable memory. Additionally,the computer-readable medium can include a magneto-optical or opticalmedium, such as a disk or tapes or other storage device to captureinformation communicated over a transmission medium. A digital fileattachment to an e-mail or other self-contained information archive orset of archives may be considered a distribution medium that isequivalent to a tangible storage medium. Accordingly, the disclosure isconsidered to include any one or more of a computer-readable medium or adistribution medium and other equivalents and successor media, in whichdata or instructions may be stored.

Although the present specification describes components and functionsthat may be implemented in particular embodiments with reference toparticular standards and protocols commonly used by investmentmanagement companies, the invention is not limited to such standards andprotocols. For example, standards for Internet and other packet switchednetwork transmission (e.g., TCP/IP, UDP/IP, HTML, HTTP) representexamples of the state of the art. Such standards are periodicallysuperseded by faster or more efficient equivalents having essentiallythe same functions. Accordingly, replacement standards and protocolshaving the same or similar functions as those disclosed herein areconsidered equivalents thereof.

According to one embodiment, systems and methods are provided forcalculating and disseminating swaption volatility indices. IRSvolatility indices (“IRS−VI”) may be calculated and disseminated usingthe systems shown in FIGS. 1, 2, and 5 and described in detail above.Generally, the IRS−VIs reflect the fair value of contracts for deliveryof volatility of a forward swap rate of arbitrary tenor, and estimatethe expected volatility of forward swap rates within arbitraryinvestment horizons. According to embodiments of the present invention,IRS−VIs can be calculated for interest rates in all currencies for whicha swaption market exists. According to an embodiment of the presentinvention, the IRS−VI is calculated based on data relating to a liquidswaption market. For example, while IRS-Vis could be calculated in anycurrency, the indices would be particularly well suited for markets thatoperate in the following currencies: USD, EUR, GBP, CHF, and JPY.

According to one embodiment of the present invention, the IRS−VIs arecalculated, for each maturity-tenor combination on the “volatilitysurface,” by aggregating the price of at-the-money and out of-the moneyreceiver and payer interest rate swaptions (i.e., the swaption “skew,”the “volatility skew”) into a single formula, which is independent ofany option pricing model. These IRS−VIs match the prevailing marketpractices to quote swaptions, for example to quote in terms of eitherbasis point volatility or percentage volatility. Moreover, the IRS−VIsdescribed herein can reflect the fair market value of contracts forfuture delivery of IRS volatility, at each point of the volatilitysurface, i.e., over any arbitrary maturity date and tenor of the forwardswap rate.

According to an embodiment of the present invention, the IRS−VIs areconstructed using the prices of a broad set of interest rate swaptions,including both at-the-money and out-of-the money swaptions. Thus,according to an embodiment of the present invention, the IRS−VIs reducethe dimensions of the “volatility cube” in the interest rate swaptionmarkets from three down to two, where a three dimensional relationshipstructure is reduced down to a two dimensional relationship structure.The “volatility cube” represents the relationship between the impliedvolatility of interest rate swaptions and: (i) time to expiration of theinterest rate swaption, (ii) the length of the tenor period of the swapunderlying the interest rate swaption, and (iii) the strike of theinterest rate swaption. It is the dimension of the “volatility cube”that represents the relationship between the implied volatility ofinterest rate swaptions and the strike of the interest rate swaptionsthat is reduced by collapsing this dimension (iii) into a single pointfor each maturity-tenor combination.

Uncertainties relating to swaptions relate to both the differencebetween the future swap rate and the fixed strike at maturity and theprice value of a basis point at maturity. Mathematically, the value ofan interest rate swap payer at maturity at time T is,

[R_(T) (T₁, . . . , T_(n))−K]×PVBP_(T) (T₁, . . . , T_(n)),   (1)

where T_(n)−T is a swap's tenor; T₁−T, . . . , T_(n)−T_(n−1) are thereset intervals of the swap contract over its tenor; R_(t) (T₁, . . . ,T_(n)) is the forward swap rate at time t; K is the fixed interest ofthe swap; and PVBP_(T) (T₁, . . . , T_(n)) is the swap's “price value ofa basis point.” According to one embodiment of the present inventionPVBP_(T) (T1, . . . , Tn) can be calculated according to the followingmathematical formula:

${{PVBP}_{T}\left( {T_{1},\ldots \mspace{14mu},T_{n}} \right)} \equiv {\overset{n}{\sum\limits_{i = 1}}{\delta_{i - 1}{P_{T}\left( T_{i} \right)}}}$

where P_(t) (T_(i)) denotes the price as of time t of a pure discountbond maturing at some future date T_(i), and δ_(i)≡T_(i+1)−T_(i).PVBP_(T) (T₁, . . . , T_(n)) is the impact of a one basis point changein a swap rate on the value of a fixed leg of the interest rate swap.

According to an embodiment of the present invention, an IRS−VI may becalculated, by the computerized financial exchange systems describedabove, as the fair value of a contract for volatility delivery, usingdata related to prices in swaption markets. According to an embodimentof the present invention, such a contract for volatility delivery may bedefined by the following parameters: whereby at time t, a counterpartypromises to pay the following payoff to a second counterparty at timeT>t:

p _(n) (t, T)≡[V _(n) (t, T)−VS _(n) (t, T)]×PVBP _(T) (T ₁ , . . . , T_(n))   (2)

where VS_(n) (t, T) is a fixed variance swap rate determined at t, andis set so as to make the time t value of this contract equal to zero;V_(n) (t, T) is the realized IRS volatility over the horizon T−t; andPVBP_(T) (T1, . . . , Tn), is as defined above.

According to an embodiment of the present invention, a forward varianceagreement may be defined by the following parameters: whereby at time t,a counterparty promises to pay the following payoff to a secondcounterparty at time T>t.

p _(n) ^(f)(t,T)≡V _(n)(t,T)×PVBP _(T)(T ₁ , . . . , T _(n))   (3)

The payoff p_(n) (t, T) in equation (2) is similar to that utilized forequity volatility contracts: the holder of the contract would receivePVBP_(T) (T₁, . . . , T_(n)) dollars for every point by which therealized variance, V_(n) (t, T), exceeds the variance strike price,VS_(n) (t, T). PVBP_(T) (T₁, . . . , T_(n)), is unknown at time t.Rescaling the rate volatility payoff by it, as done in Eq. (2), ismathematically unavoidable, when the objective is to price IRSvolatility in a way independent of any option pricing model.Mathematically, to price volatility in IRS markets, simultaneously withother fixed income instruments such as swaps, swaptions and purediscount bonds, the payoff of the variance contract in equation (2)needs to be rescaled by PVBP_(T) (T₁, . . . , T_(n)), just as the valueof the swap does in equation (1).

According to two embodiments of the present invention, V_(n) (t, T) canbe calculated using two different algorithms, which give rise to twodifferent and distinct sets of contracts and indices, which match twodistinct strands of market practice to quote interest rate swaptions. Afirst strand is based on quoting swaptions in terms of basis pointimplied volatilities (which are percentage implied volatilities forBlack's formula (Black, Fisher, “The Pricing of Commodity Contracts,”Journal of Financial Economics 3, 167-179 (1976), multiplied by thecurrent forward swap rate). A second strand is based on swaptions quotedin terms of percentage implied volatilities.

According to one embodiment of the present invention, a basis pointIRS−VI, as mathematically defined in equation (6) below, aggregatesinformation of basis point implied volatilities by tracking expectedvolatility in a risk-adjusted market cast in basis point terms. In anembodiment of the present invention, the formula used to calculate IRSrealized volatility gives rise to the so called “basis point” or“Gaussian” realized variance, where:

V _(n)(t,T)≡∫_(t) ^(T) R _(s) ²(T ₁ , . . . , T _(n))∥σ_(s)(T ₁ , . . ., T _(n))∥² ds   (4)

where R_(s)(T₁, . . . , T_(n)) is the forward swap rate at time s withfixed payments on T₁, . . . , T_(n) which is a diffusion process withstochastic volatility and its arithmetic changes can be expressed as

dR _(s)(T ₁ , . . . , T _(n))=R _(s)(T ₁ , . . . , T _(n))σ_(s)(T ₁ , .. . , T _(n))dW _(s) * , s∈[t,T]

where σ_(s)(T₁, . . . , T_(n)) is adapted to W_(s)*, which is amultidimensional Brownian motion under the swap probability measuredefined by

${\frac{{dQ}_{swap}}{dQ}}_{F_{T}} = {{\exp \left( {- {\int_{t}^{T}{r_{s}{ds}}}} \right)}\frac{{PVBP}_{T}\left( {T_{1},\ldots \mspace{14mu},T_{n}} \right)}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{14mu},T_{n}} \right)}}$

where Q is the risk neutral probability measure, F_(T) is theinformation set at time T, and r_(s) is the instantaneous interest rateat time s.

-   Given this definition, the fair value of VS_(n) (t, T) in equation 2    equals:

VS _(n)(t,T)=(T−t)×[IRS−VI _(n) ^(BP)(t,T)]²   (5)

where IRS−VI_(n) ^(BP) (t, T) is an interest rate swap volatility indexaccording to one embodiment of the present invention. This embodiment ofthe present invention may be referred to as a Basis Point IRS−VI orBPIRS−VI. This index is calculated according to the followingmathematical formula:

$\begin{matrix}{{{IRS} - {{VI}_{n}^{BP}\left( {t,T} \right)}} \equiv \sqrt{\frac{1}{T - t}{\frac{2}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{14mu},T_{n}} \right)}\begin{bmatrix}{{\sum\limits_{i:{K_{i} < R_{t}}}{{{SWPN}_{t\;}^{R}\left( {K_{i},{T;T_{n}}} \right)}^{\prime}\Delta \; K_{i}}} +} \\{\sum\limits_{i:{K_{i} \geq R_{t}}}{{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}^{\prime}\Delta \; K_{i}}}\end{bmatrix}}}} & (6)\end{matrix}$

where SWPN_(t) ^(R)(K_(i),T;T_(n)) (resp., SWPN_(t) ^(P)(K_(i),T;T_(n)))is the price of a swaption receiver (resp., payer), struck at K_(i),expiring at T and with tenor extending up to time T_(n), andΔK_(i)=1/2(K_(i+1)−K_(i−1)) for i≧1, ΔK₀=(K₁−K₀),ΔK_(M)=(K_(M)−K_(M−1)), where K₀ and K_(M) are the lowest and thehighest available strike prices traded in the market, and M+1 is thetotal number of traded swaptions expiring at time T and with tenorextending up to time T_(n).

Pricing volatility in swap markets, in a way independent of any optionpricing model, entails a contract design where two components need to besimultaneously taken into account: one, related to the realized varianceof the forward swap rate, and a second, related to the forward PVBP, asthe payoff in equation (1) indicates.

According to an embodiment of the present invention, an algorithm tocompute the IRS variance Vn (t, T), alternative to Eq. (4), relies onthe “log” or “percentage” realized variance:

V _(n)(t,T)≡∫_(t) ^(T)∥σ_(s)(T ₁ , . . . , T _(n))∥² ds   (7)

where the log changes of the forward swap rate at time s with fixedpayments on T₁, . . . , T_(n), R_(s)(T₁, . . . , T_(n)), which is adiffusion process with stochastic volatility, can be expressed as

d ln R _(s)(T ₁ , . . . , T _(n))=−1/2∥σ_(s)(T ₁ , . . . , T _(n)∥² ds+σ_(s)(T ₁ , . . . , T _(n))·dW _(s) *, s∈[t,T]

where σ_(s)(T₁, . . . , T_(n)) is adapted to W_(s)*, which is amultidimensional Brownian motion under the swap probability measuredefined by

${\frac{{dQ}_{swap}}{dQ}_{F_{T}}} = {{\exp \left( {- {\int_{t}^{T}{r_{s}{ds}}}} \right)}\frac{{PVBP}_{T}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}}$

where Q is the risk neutral probability measure, F_(T) is theinformation set at time T, and r_(s) is the instantaneous interest rateat time s.

-   The fair value of VS_(n) (t, T), relating to the contract delivering    p_(n) (t, T) in Eq. (2), equals:

VS _(n) (t, T)=(T−t)×[IRS−VI _(n) (t, T)]²   (8)

and the fair value for delivery of p_(n) ^(f) (t,T) in equation (3) is:

V−Fwd _(n) (t, T)≡PVBP _(t) (T ₁ , . . . , T _(n))·VS _(n) (t, T)   (9)

where IRS−VI_(n) (t, T) is an embodiment of a value of an interest rateswap index according to the present invention. This embodiment of thepresent invention may be referred to as a percentage IRS−VI. Accordingto an embodiment of the present invention, the Percentage IRS−VI may becalculated as follows:

$\begin{matrix}{{{IRS} - {{VI}_{n}\left( {t,T} \right)}} \equiv \sqrt{\frac{1}{T - t}{\frac{2}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}\begin{bmatrix}{{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} < R}{\frac{{SWPN}_{t}^{R}\left( {K_{i},{T;T_{n}}} \right)}{K_{i}^{2}}\Delta \; K_{i}}} -} \\{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} \geq R}{\frac{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}{K_{i}^{2}}\Delta \; K_{i}}}\end{bmatrix}}}} & (10)\end{matrix}$

where SWPN_(t) ^(R) (K_(i),T;T_(n)) (resp., SWPN_(t) ^(P)(K_(i),T;T_(n))) is the price of a swaption receiver (resp., payer),struck at K_(i), expiring at T and with tenor extending up to timeT_(n), and ΔK_(i)=1/2(K _(i+1)−K_(i−1)) for i≧1, ΔK₀=(K₁−K₀),ΔK_(M)=(K_(M)−K_(M−1)), where K₀ and K_(M) are the lowest and thehighest available strike prices traded in the market, and M+1 is thetotal number of traded swaptions expiring at time T and with tenorextending up to time T_(n).

The difference between the Basis Point IRS−VI and the Percentage IRS−VIis that IRS−VI_(n) (t, T), when calculated according to the PercentageIRS−VI, aggregates information conveyed by percentage impliedvolatilities from at/out-of-the money swaptions, not basis pointvolatilities.

According to another embodiment of the present invention, the BasisPoint IRS−VI may be calculated as follows:

$\begin{matrix}{{{IRS} - {{VI}_{n}^{BP}\left( {t,T} \right)}} \equiv \sqrt{\frac{2}{T - t}\begin{bmatrix}{{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} < R}{\frac{1}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}{{SWPN}_{t}^{R}\left( {K_{i},{T;T_{n}}} \right)}\Delta \; K_{i}}} +} \\{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} \geq R}{\frac{1}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}\Delta \; K_{i}}}\end{bmatrix}}} & (11)\end{matrix}$

where SWPN_(t) ^(R) (K_(i),T;T_(n)) (resp., SWPAN_(t) ^(P)(K_(i),T;T_(n))) is the price of a swaption receiver (resp., payer),struck at K_(i), expiring at T and with tenor extending up to timeT_(n), and ΔK_(i)=1/2(K_(i+1)−K_(i−1)) for i≧1, ΔK₀=(K₁−K₀),ΔK_(M)=(K_(M)−K_(M−1)), where K₀ and K_(M) are the lowest and thehighest available strike prices traded in the market, and M+1 is thetotal number of traded swaptions expiring at time T and with tenorextending up to time T_(n).

According to yet another embodiment of the present invention, thePercentage IRS−VI may be calculated as follows:

$\begin{matrix}{{{IRS} - {{VI}_{n}^{BP}\left( {t,T} \right)}} \equiv \sqrt{\frac{2}{T - t}\begin{bmatrix}{{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} < R}{{{SWPN}_{t}^{R}\left( {K_{i},{T;T_{n}}} \right)}^{\prime}\Delta \; K_{i}}} +} \\{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} \geq R}{{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}^{\prime}\Delta \; K_{i}}}\end{bmatrix}}} & (12)\end{matrix}$

where SWPN_(t) ^(R) (K_(i),T;T_(n))′ (resp., SWPN_(t) ^(P)(K_(i),T;T_(n))′) is equal to

$\frac{{SWPN}_{t}^{R}\left( {K_{i},{T;T}} \right)}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}$

(resp.,

$\left. \frac{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)} \right)$

where SWPN_(t) ^(R) (K_(i),T;T_(n)) (resp., SWPN_(t) ^(P)(K_(i),T;T_(n))) is the price of a swaption receiver (resp., payer),struck at Ki, expiring at T and with tenor extending up to time T_(n),and ΔK_(i)=1/2(K_(i+1)−K_(i−1)) for i≧1, ΔK₀=(K₁−K₀),ΔK_(M)=(K_(M)−K_(M−1)), where K₀ and K_(M) are the lowest and thehighest available strike prices traded in the market, and M+1 is thetotal number of traded swaptions expiring at time T and with tenorextending up to time Tn. According to an embodiment of the presentinvention SWPN_(t) ^(R) (K_(i),T;T_(n))′ and SWPN_(t) ^(P)(K_(i),T;T_(n))′ can be calculated using quoted implied volatilities ofthe swaptions in Black's formula. In other embodiments, the basis pointcalculation method of equation (11) or the percentage calculation methodof equation (12) may utilize a mix of both implied volatility data andprice data, rather than only one of these types of data, as input datafor calculating the interest rate swap volatility index.

According to another embodiment of the present invention, the PercentageIRS−VI may be calculated as follows:

$\begin{matrix}{{{IRS} - {{VI}_{n}\left( {t,T} \right)}} \equiv \sqrt{\frac{2}{T - t}\begin{bmatrix}{{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} < R}{\frac{1}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}\frac{{SWPN}_{t}^{R}\left( {K_{i},{T;T_{n}}} \right)}{K_{i}^{2}}\Delta \; K_{i}}} +} \\{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} \geq R}{\frac{1}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}\frac{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}{K_{i}^{2}}\Delta \; K_{i}}}\end{bmatrix}}} & (13)\end{matrix}$

where SWPN_(t) ^(R) (K_(i),T;T_(n)) (resp., SWPN_(t) ^(P)(K_(i),T;T_(n))) is the price of a swaption receiver (resp., payer),struck at K_(i), expiring at T and with tenor extending up to timeT_(n), and ΔK_(i)=1/2(K_(i+1)−K_(i−1)) for i≧1, ΔK₀=(K₁−K₀),ΔK_(M)=(K_(M)−K_(M−1)), where K₀ and K_(M) are the lowest and thehighest available strike prices traded in the market, and M+1 is thetotal number of traded swaptions expiring at time T and with tenorextending up to time T_(n).

According to yet another embodiment of the present invention, thePercentage IRS−VI may be calculated as follows:

$\begin{matrix}{{{IRS} - {{VI}_{n}\left( {t,T} \right)}} \equiv \sqrt{\frac{2}{T - t}\begin{bmatrix}{{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} < R}{\frac{{{SWPN}_{t}^{R}\left( {K_{i},{T;T_{n}}} \right)}^{\prime}}{K_{i}^{2}}\Delta \; K_{i}}} +} \\{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} \geq R}{\frac{{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}^{\prime}}{K_{i}^{2}}\Delta \; K_{i}}}\end{bmatrix}}} & (14)\end{matrix}$

where SWPN_(t) ^(R) (K_(i),T;T_(n))′ (resp., SWPN_(t) ^(P)(K_(i),T;T_(n))′) is equal to

$\frac{{SWPN}_{t}^{R}\left( {K_{i},{T;T}} \right)}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)}$

(resp.,

$\left. \frac{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{11mu},T_{n}} \right)} \right)$

where SWPN_(t) ^(R) (K_(i),T;T_(n)) (resp., SWPN_(t) ^(P)(K_(i),T;T_(n))) is the price of a swaption receiver (resp., payer),struck at Ki, expiring at T and with tenor extending up to time T_(n),and ΔK_(i)=1/2(K_(i+1)−K_(i−1)) for i≧1, ΔK₀=(K₁−K₀),ΔK_(M)=(K_(M)−K_(M−1)), where K₀ and K_(M) are the lowest and thehighest available strike prices traded in the market, and M+1 is thetotal number of traded swaptions expiring at time T and with tenorextending up to time Tn. According to an embodiment of the presentinvention SWPN_(t) ^(R) (K_(i),T;T_(n))′ and SWPN_(t) ^(P)(K_(i),T;T_(n))′ can be calculated using quoted implied volatilities ofthe swaptions in Black's formula. In other embodiments, the basis pointcalculation method of equation (13) or the percentage calculation methodof equation (14) may utilize a mix of both implied volatility data andprice data, rather than only one of these types of data, as input datafor calculating the interest rate swap volatility index.

FIG. 3, is a flow diagram that outlines an embodiment of the steps forcalculating and disseminating a basis point interest rate swapvolatility index according to the present invention. At step 302, datais received electronically from an electronic data source. Included inthe received data is data regarding the maturity, tenor, strike, andimplied volatility of interest rate swaptions. At step 304, the data iscleaned and normalized, according to known techniques, and a swaptionimplied volatility cube is created in order to generate the impliedvolatilities for all available maturity/tenor/strike combinations. Atstep 306 the implied volatilities are converted to prices using Black'sformula. At step 308, the prices for each maturity and tenor combinationfor all available strikes are inputted into equation (6), shown above,to calculate a basis point interest rate swap volatility index.

FIG. 4, is a flow diagram that outlines an embodiment of the steps forcalculating and disseminating a percentage interest rate swap volatilityindex according to the present invention. At step 402, data is receivedelectronically from an electronic data source. Included in the receiveddata is data regarding the maturity, tenor, strike, and impliedvolatility of interest rate swaptions. At step 404, the data is cleanedand normalized, according to known techniques, and a swaption impliedvolatility cube is created in order to generate the implied volatilitiesfor all available maturity/tenor/strike combinations. At step 406 theimplied volatilities are converted to prices using Black's formula. Atstep 408, the prices for each maturity and tenor combination for allavailable strikes are inputted into equation (10), shown above, tocalculate a percentage interest rate swap volatility index.

The steps shown in FIGS. 3 and 4 can be performed using the systemsillustrated in FIGS. 1, 2, and 5.

IMPLEMENTATION EXAMPLES

The following is a non-limiting example of how the methodologies of thepresent invention can be used to construct the Basis Point IRS−VI andthe Percentage IRS−VI. As noted above the actual calculation anddissemination of the Basis Point IRS−VI and the Percentage IRS−VI areperformed by the calculation and dissemination system, an example ofwhich is illustrated in FIG. 3.

The present example, utilizes data reflecting market conditions on Feb.12, 2010. The data provided are implied volatilities expressed inpercentage terms, and relate to interest rate swaptions maturing in onemonth and tenor equal to five years. The data for this example isprovided below in table 1:

TABLE 1 Basis Point Black's prices Strike Percentage Implied ReceiverPayer Rate (%) Implied Vol Vol Swaption ({circumflex over (Z)}) SwaptionZ 1.7352 36.1900 98.9869 ≈0 10.0000 · 10⁻³  1.9852 36.1900 98.98690.0007 · 10⁻³ 7.5007 · 10⁻³ 2.2352 36.1200 98.7954 0.0259 · 10⁻³ 5.0259· 10⁻³ 2.4352 35.9900 98.4398 0.1773 · 10⁻³ 3.1773 · 10⁻³ 2.5352 35.930098.2757 0.3692 · 10⁻³ 2.3692 · 10⁻³ 2.6352 35.8600 98.0843 0.6793 · 10⁻³1.6793 · 10⁻³ 2.6852 35.8300 98.0022 0.8855 · 10⁻³ 1.3855 · 10⁻³ 2.735235.8000 97.9202 1.1272 · 10⁻³ 1.1272 · 10⁻³ 2.7852 35.7600 97.81081.4037 · 10⁻³ 0.9037 · 10⁻³ 2.8352 35.7300 97.7287 1.7142 · 10⁻³ 0.7142· 10⁻³ 2.9352 35.6700 97.5646 2.4270 · 10⁻³ 0.4270 · 10⁻³ 3.0352 35.600097.3731 3.2406 · 10⁻³ 0.2406 · 10⁻³ 3.2352 35.4700 97.0175 5.0644 · 10⁻³0.0644 · 10⁻³ 3.4852 35.3100 96.5799 7.5092 · 10⁻³ 0.0092 · 10⁻³ 3.735235.1400 96.1149 10.0010 · 10⁻³  0.0010 · 10⁻³The first two columns of Table 1, as shown above, report strike rates,K, and percentage implied volatilities for each strike rate, IV(K). Thethird column provides basis point implied volatilities, IV^(BP) (K), foreach strike rate K, defined according to the following formula:IV^(BP)(K)=IV(K)·R, where R denotes the current forward swap rate.According to other embodiments of the present invention, basis pointvolatilities are not needed to compute the indexes of the presentinvention, such as Basis Point IRS−VI and Percentage IRS−VI.

According to an embodiment of the present invention, the Basis PointIRS−VI and Percentage IRS−VI are calculated by first, plugging the“skew” IV(K) into the Black's formula, and then, replacing the Black'sformula into the formulas shown above for calculating the Basis PointIRS−VI and Percentage IRS−VI, e.g., equations (6) and (10). Accordingly:

$\begin{matrix}{{{IRS} - {{VI}_{n}^{BP}\left( {t,T} \right)}} = \sqrt{\frac{2}{T - t}\begin{bmatrix}{{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} < R_{t}}{{\hat{Z}\left( {R,T,{K_{i};{\left( {T - t} \right){{IV}^{2}\left( K_{i} \right)}}}} \right)}\Delta \; K_{i}}} +} \\{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} \geq R_{t}}{{Z\left( {R,T,{K_{i};{\left( {T - t} \right){{IV}^{2}\left( K_{i} \right)}}}} \right)}\Delta \; K_{i}}}\end{bmatrix}}} & (15) \\{\mspace{79mu} {and}} & \; \\{{{IRS} - {{VI}_{n}\left( {t,T} \right)}} = \sqrt{\frac{2}{T - t}\begin{bmatrix}{{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} < R_{t}}{\frac{\hat{Z}\left( {R,T,{K_{i};{\left( {T - t} \right){{IV}^{2}\left( K_{i} \right)}}}} \right)}{K_{i}^{2}}\Delta \; K_{i}}} +} \\{\sum\limits_{{i\text{:}\mspace{14mu} K_{i}} \geq R_{t}}\frac{{Z\left( {R,T,{K_{i};{\left( {T - t} \right){{IV}^{2}\left( K_{i} \right)}}}} \right)}\Delta \; K_{i}}{K_{i}^{2}}}\end{bmatrix}}} & (16) \\{\mspace{79mu} {where}} & \; \\{{\hat{Z}\left( {R,T,{K_{i};{\left( {T - t} \right){{IV}^{2}\left( K_{i} \right)}}}} \right)} = {{Z\left( {R,T,{K_{i};{\left( {T - t} \right){{IV}^{2}\left( K_{i} \right)}}}} \right)} + K - R}} & (17) \\{\mspace{79mu} {{{Z\left( {R,T,{K;V}} \right)} = {{R\; {\Phi (d)}} - {K\; {\Phi \left( {d - \sqrt{V}} \right)}}}},{d = \frac{{\ln \frac{R}{K}} + {\frac{1}{2}V}}{\sqrt{V}}}}} & (18)\end{matrix}$

and φ denotes the cumulative standard normal distribution.

According to the present example, the percentage implied volatilities,IV(K), are utilized as data in equations 17 and 18, to obtain values for{circumflex over (Z)} and Z. The fourth and fifth columns of Table 1, asshown above, provide interest rate swaption prices re-normalized by thePVBP, i.e., the values of {circumflex over (Z)} and Z (Black's prices),for each strike rate.

Table 2, as shown below, provides information regarding the presentexamples calculation of the Basis Point IRS−VI and Percentage IRS−VI,according to equations 15 and 16 respectively.

TABLE 2 Weights Contributions to Strikes Strike Swaption Basis PointPercentage Basis Point Percentage Rate (%) Type Price ΔK_(i)ΔK_(i)/K_(i) ² Contribution Contribution 1.7352 Receiver ≈0 0.00256.3031 ≈0 ≈0 1.9352 Receiver 0.0007 · 10⁻³ 0.0025 6.3435 0.0018 · 10⁻⁸0.0046 · 10⁻³ 3.2352 Receiver 0.0259 · 10⁻³ 0.0022 4.5035 0.0583 · 10⁻⁸0.1167 · 10⁻³ 3.4352 Receiver 0.1773 · 10⁻³ 0.0015 2.5294 0.2660 · 10⁻⁸0.4485 · 10⁻³ 2.5352 Receiver 0.3692 · 10⁻³ 0.0010 1.5559 0.3692 · 10⁻⁸0.3744 · 10⁻³ 2.6352 Receiver 0.6793 · 10⁻³ 0.0008 1.0800 0.5095 · 10⁻⁸0.7337 · 10⁻³ 2.6652 Receiver 0.8855 · 10⁻³ 0.0005 0.6935 0.4428 · 10⁻⁸0.6141 · 10⁻³ 2.7352 ATM 1.1272 · 10⁻³ 0.0005 0.6683 0.5636 · 10⁻⁸0.7533 · 10⁻³ 2.7352 Payer 0.9037 · 10⁻³ 0.0005 0.6446 0.4518 · 10⁻⁸0.5325 · 10⁻³ 2.8352 Payer 0.7142 · 10⁻³ 0.0007 0.9330 0.5357 · 10⁻⁸0.6664 · 10⁻³ 2.9352 Payer 0.4370 · 10⁻³ 0.0010 1.1607 0.4370 · 10⁻⁸0.4956 · 10⁻³ 3.0352 Payer 0.2406 · 10⁻³ 0.0015 1.6282 0.3609 · 10⁻⁸0.3917 · 10⁻³ 3.2352 Payer 0.0644 · 10⁻³ 0.0023 2.1497 0.1448 · 10⁻⁸0.1384 · 10⁻³ 3.4852 Payer 0.0092 · 10⁻³ 0.0026 2.0082 0.0229 · 10⁻⁸0.0188 · 10⁻³ 3.7352 Payer 0.0010 · 10⁻³ 0.0026 1.7919 0.0024 · 10⁻⁸0.0017 · 10⁻³ SUMS 4.1567 · 10⁻⁸ 0.5405 · 10⁻³

The second column of Table 2 displays the type of out-of-the moneyinterest rate swaptions entering in the calculations of the embodimentsof the IRS−VI. The third column money swaption entering into thecalculation; the third column has the Black's price corresponding to theused interest rate swaption; the fourth and fifth columns report theweights each Black's price bears towards the final computation of theindex, before the final rescaling of

$\frac{2}{T - t};$

and finally, the sixth and seventh columns report each out-of-the moneyinterest rate swaption price corrected by the appropriate weight. Eachprice in the third column is multiplied by the corresponding weight inthe fourth column, for the “Basis Point Contribution,” and each price inthe third column is multiplied by the corresponding weight in the fifthcolumn, for the “Percentage Contribution.”

Thus, according to the data provided in this example, embodiments of theBasis Point IRS−VI and Percentage IRS−VI are calculated, respectively,as follows:

${{{IRS} - {VI}^{BP}} = {{100^{2}\sqrt{{\frac{2}{12^{- 1}}4.1567} \cdot 10^{- 6}}} = 99.8803}},{and}$${{IRS} - {VI}} = {{100\sqrt{{\frac{2}{12^{- 1}}5.5405} \cdot 10^{- 3}}} = {36.4653.}}$

For purposes of comparison, the at-the-money implied basis point andpercentage volatilities are IV^(BP) (R)=97.9202 and IV(R)=35.8000.

In this non-limiting example, the basis point index is rescaled by 100²,to mimic the market practice to express basis point implied volatilityas the product of rates times log-volatility, where both rates andlog-volatility are multiplied by 100.

According to embodiments of the present invention, indices calculatedaccording to the embodiments of the present invention may serve as theunderlying asset for derivative contracts, such as options and futurescontracts. More particularly, according to an embodiment of the presentinvention, an IRS−VI may serve as the underlying reference forderivative contracts designed for trading the volatility of forwardinterest rate swap rates of various tenors. In particular, futures andoptions contracts with varying maturities based on the index may betraded OTC and/or listed on exchanges.

Derivative instruments based on the interest rate swaption volatilityindex disclosed above may be created as standardized, exchange-tradedcontracts, as opposed to over-the-counter contracts. Once the interestrate swaption volatility index (IRS−VI) based on interest rate swaptionsis calculated, the index may be accessed for use in creating aderivative contract, and the derivative contract may be assigned aunique symbol. Generally, the IRS−VI derivative contract may be assignedany unique symbol that serves as a standard identifier for the type ofstandardized IRS−VI derivative contract. Information associated with theIRS−VI and/or the IRS−VI derivative contract may be transmitted fordisplay, such as transmitting information to list the IRS−VI indexand/or the IRS−VI derivative on a trading platform. Examples of thetypes of information that may be transmitted for display include asettlement price of an IRS−VI derivative, a bid or offer associated withan IRS−VI derivative, a value of an IRS−VI index, and/or a value of anunderlying swaptions that an IRS−VI is associated with.

Generally, an IRS−VI derivative contract may be listed on an electronicplatform, an open outcry platform, a hybrid environment that combinesthe electronic platform and open outcry platform, or any other type ofplatform known in the art. One example of a hybrid exchange environmentis disclosed in U.S. Pat. No. 7,613,650, filed Apr. 24, 2003, theentirety of which is herein incorporated by reference. Additionally, atrading platform such as an exchange may transmit IRS−VI derivativecontract quotes of liquidity providers over dissemination networks toother market participants. Liquidity providers may include DesignatedPrimary Market Makers (“DPM”), market makers, locals, specialists,trading privilege holders, registered traders, members, or any otherentity that may provide a trading platform with a quote for a variancederivative. Dissemination Networks may include networks such as theOptions Price Reporting Authority (“OPRA”), the CBOE Futures Network, anInternet website or email alerts via email communication networks.Market participants may include liquidity providers, brokerage firms,normal investors, or any other entity that subscribes to a disseminationnetwork.

The trading platform may execute buy and sell orders for the IRS−VIderivative and may repeat the steps of calculating the IRS−VI of theunderlying swaptions, accessing the IRS−VI index, transmittinginformation for the IRS−VI index and/or the IRS−VI derivative fordisplay (list the IRS−VI and/or IRS−VI derivative on a tradingplatform), disseminating the IRS−VI and/or the IRS−VI derivative over adissemination network, and executing buy and sell orders for the IRS−VIderivative until the IRS−VI derivative contract is settled.

In some implementations, IRS−VI derivative contracts may be tradedthrough an exchange-operated parimutuel auction and cash-settled basedon the IRS−VI index of log returns of the underlying equity. Anelectronic parimutuel, or Dutch, auction system conducts periodicauctions, with all contracts that settle in-the-money funded by thepremiums collected for those that settle out-of-the-money.

As mentioned, in a parimutuel auction, all the contracts that settlein-the-money are funded by those that settle out-of-the-money. Thus, thenet exposure of the system is zero once the auction process iscompleted, and there is no accumulation of open interest over time.Additionally, the pricing of contracts in a parimutuel auction dependson relative demand; the more popular the strike, the greater its value.In other words, a parimutuel action does not depend on market makers toset a price; instead the price is continuously adjusted to reflect thestream of orders coming into the auction. Typically, as each orderenters the system, it affects not only the price of the sought-afterstrike, but also affects all the other strikes available in thatauction. In such a scenario, as the price rises for the moresought-after strikes, the system adjusts the prices downward for theless popular strikes. Further, the process does not require the matchingof specific buy orders against specific sell orders, as in manytraditional markets. Instead, all buy and sell orders enter a singlepool of liquidity, and each order can provide liquidity for other ordersat different strike prices and the liquidity is maintained such thatsystem exposure remains zero. This format maximizes liquidity, a keyfeature when there is no tradable underlying instrument.

The following characteristics of futures contracts illustrate oneembodiment of a futures contract having an index of the presentinvention as an underlying asset. The characteristics are not meant tolimit the present invention, but rather to set forth commoncharacteristics of futures.

Contract Size: The notional amount of one unit of the contract may bedefined as a multiple of the index level, which may depend on thecurrency of the underlying index. When traded OTC, the multiplier may benegotiated between the parties involved on a trade-by-trade basis.

Contract Months: An exchange may list contracts with a pre-determinedsequence of maturity dates, e.g. the 3rd Friday of each of the next 6months. Similarly, OTC dealers may make markets in a pre-determinedsequence of maturity dates but may also make markets for contracts thatmature on other dates on a trade-by-trade basis.

Quotation & Minimum Price Intervals: Futures based on the index may bequoted in points and decimals or fractions that represent some notionalamount per contract and there may be a minimum increment by which thepricing of the contracts may vary, both of which may depend on thecurrency of the underlying index. The OTC market may adopt differentconventions for quoting and minimum ticks.

Last Trading Date: For each contract, a last trading date will bespecified.

Final Settlement Date: For each contract, a final settlement date willbe specified.

Final Settlement Value: The final settlement value shall be based on thelevel of the index computed at a pre-specified time on the settlementdate.

Delivery: Settlement of futures based on the index will take the form ofa delivery of the cash settlement amount and a payment date will bespecified in relation to the final settlement date.

Additional Specifications when Exchange Traded: When traded on anexchange, trading platform, margin requirements, trading hours, ordercrossing rules, block trading rules, reporting rules, and other detailsmay be specified.

The following characteristics of options contracts illustrate oneembodiment of an options contract having an index of the presentinvention as an underlying asset. The characteristics are not meant tolimit the present invention, but rather to set forth commoncharacteristics of options.:

Contract Size: The notional amount of one unit of the contract may bedefined as a multiple of the index level, which may depend on thecurrency of the underlying index. When traded OTC, the multiplier may benegotiated between the parties involved on a trade-by-trade basis.

Contract Months: An exchange may list contracts with a pre-determinedsequence of expiration dates, e.g. the 3rd Friday of each of the next 6months. Similarly, OTC dealers may make markets in a pre-determinedsequence of maturity dates but may also make markets for contracts thatexpire on other dates on a trade-by-trade basis.

Strike Prices: For each currency, strike prices that are in-, at-, andout-of the money may be listed by an exchange or quoted by OTC dealersand new strike prices may be traded as swap rates increase and decrease.An exchange or the OTC dealer community may fix a minimum incrementbetween strike prices, depending on the currency of the underlyingindex.

Quotation & Minimum Price Intervals: Options based on the index may bequoted in points and decimals or fractions that represent some notionalamount per contract and there may be a minimum increment by which thepricing of the contracts may vary, both of which may depend on thecurrency of the underlying index. The OTC Market may adopt differentconventions for quoting and minimum ticks.

Exercise Style: Options written on the IRS−VI are likely to be, but notlimited to, European style. It is envisioned that American stylecontracts could also have an index of the present invention as anunderlying asset

Expiration Date: For each contract, an expiration date will bespecified.

Last Trading Date: For each contract, a last trading date will bespecified.

Settlement of Exercise: The final settlement value shall be based on thelevel of the index computed at a pre-specified time on the settlementdate. The cash settlement amount will be the difference between theindex level and the strike price, possibly adjusted by some multiplier,and a payment date will be specified in relation to the expiration date.

Additional Specifications when Exchange Traded: When traded on anexchange, trading platform, margin requirements, trading hours,reporting rules, and other details may be specified.

According to other embodiments of the present invention, other financialproducts that track or reference the indices of the present inventionmay be created. Such products include, but are not limited to, ExchangeTraded Funds and Exchange Traded Notes listed on exchanges andstructured products sold by financial institutions.

Interest rate swap volatility indexes and derivative instruments basedthere on have been disclosed. An advantage of derivatives based on theinterest rate swap volatility indexes disclosed herein is the ability toprovide a hedge against options or other derivatives that are subject tointerest rate swap volatility risk. It is intended that the foregoingdetailed description be regarded as illustrative rather than limiting,and that it be understood that it is the following claims, including allequivalents, that are intended to define the spirit and scope of thisinvention.

1. A computer-implemented method of calculating an interest rate swapvolatility index, the method comprising: with a processor in a tradingplatform: calculating an interest rate swap volatility index associatedwith at least one underlying interest rate swaption; wherein theinterest rate swap volatility index is calculated according to theequation:${{IRS} - {{VI}_{n}^{BP}\left( {t,T} \right)}} \equiv \sqrt{\frac{2}{T - t}\left\lbrack {{\sum\limits_{i:{K_{i} < R}}{{{SWPN}_{t\;}^{R}\left( {K_{i},{T;T_{n}}} \right)}^{\prime}\Delta \; K_{i}}} + {\sum\limits_{i:{K_{i} \geq R}}{{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}^{\prime}\Delta \; K_{i}}}} \right\rbrack}$where SWPN_(t) ^(R) (K_(i),T;T_(n))′ (resp., SWPN_(t) ^(P)(K_(i),T;T_(n))′) is equal to SWPN_(t) ^(R) (K_(i),T;T)/PVBP_(t) (T₁, .. . , T_(n)) (resp.,$\left. \frac{{SWPN}_{t}^{P}\left( {K_{i},{T;T_{n}}} \right)}{{PVBP}_{t}\left( {T_{1},\ldots \mspace{14mu},T_{n}} \right)} \right)$where SWPN_(t) ^(R) (K_(i),T;T_(n)) (resp., SWPN_(t) ^(P)(K_(i),T;T_(n))) is a price of a swaption receiver (resp., payer),struck at Ki, expiring at T and with tenor extending up to time T_(n),and ΔK_(i)=1/2(K_(i+1)−K_(i−1)) for i≧1, ΔK₀=(K₁−K₀),ΔK_(M)=(K_(M)−K_(M−1)), where K₀ and K_(M) are lowest and highestavailable strike prices traded in a market, and M+1 is a total number oftraded swaptions expiring at time T and with tenor extending up to timeTn, where SWPN_(t) ^(R) (K_(i),T;T_(n))′ and SWPN_(t) ^(P)(K_(i),T;T_(n))′ are calculated using quoted implied volatilities of theat least one underlying interest rate swaption in Black's formula, andwhere PVBPt (T₁, . . . , T_(n)) is a price value of a basis point attime t of an interest rate swap starting at time T with fixed paymentdates T₁, . . . , T_(n), which is an impact of a one basis point changein a swap rate on a value of a fixed leg of the interest rate swap;displaying the interest rate swap volatility index associated with theunderlying interest rate swaption on a trading platform display devicecoupled with the trading platform; creating a standardizedexchange-traded derivative instrument based on the interest rate swapvolatility index; and transmitting information for display associatedwith the standardized exchange traded derivative instrument, theinformation comprising a settlement price of the derivative instrument.2. The method of claim 1, wherein calculating the interest rate swapvolatility index comprises the processor aggregating prices ofat-the-money and out-of-the-money receiver and payer interest rateswaptions.
 3. The method of claim 1, wherein calculating the interestrate swap volatility index comprises the processor aggregating impliedvolatilities of at-the-money and out-of-the-money receiver and payerinterest rate swaptions.
 4. The method of claim 2, wherein calculatingthe interest rate swap volatility index further comprises the processoraggregating the prices in a single equation that is independent of anyoption pricing model.
 5. The method of claim 3, wherein calculating theinterest rate swap volatility index further comprises the processorconverting the implied volatilities to prices and aggregating the pricesin a single equation that is independent of any option pricing model. 6.The method of claim 4, wherein the processor reduces a relationshipbetween prices of the underlying interest rate swaptions and theircorresponding strikes into a single measure of volatility for eachmaturity-tenor combination of the underlying interest rate swaptions. 7.The method of claim 5, wherein the processor reduces a relationshipbetween implied volatilities of the underlying interest rate swaptionsand their corresponding strikes into a single measure of volatility foreach maturity-tenor combination of the underlying interest rateswaptions. 8.-11. (canceled)
 12. The method of claim 1, furthercomprising: transmitting the interest rate swap volatility indexassociated with the underlying interest rate swaption over at least oneelectronic dissemination network.
 13. (canceled)
 14. The method of claim1, wherein transmitting information associated with the derivativeinstrument based on the interest rate swap volatility index comprisestransmitting a settlement price of the derivative instrument.
 15. Thetrading platform of claim 1, wherein transmitting information associatedwith the derivative instrument based on the interest rate swapvolatility index comprises transmitting a bid associated the derivativeinstrument.
 16. The trading platform of claim 1, wherein transmittinginformation associated with the derivative instrument based on theinterest rate swap volatility index comprises transmitting an offerassociated the derivative instrument.
 17. A trading platform comprising:a display device; a memory storing a set of instructions for calculatingan interest rate swap volatility index associated with an underlyinginterest rate swaption; and a processor in communication with thedisplay device and the memory, the processor configured to execute theset of instructions stored in the memory and to: calculate the interestrate swap volatility index associated with the underlying interest rateswaption, wherein to calculate the interest rate swap volatility index,the processor is configured to aggregate implied volatilities ofat-the-money and out-of-the-money receiver and payer interest rateswaptions; display the interest rate swap volatility index associatedwith the underlying interest rate swaption on the display device;transmit the interest rate swap volatility index associated with theunderlying interest rate swaption over at least one electronicdissemination network; wherein the processor is further configured tocalculate the interest rate swap volatility index by converting impliedvolatilities to prices and aggregating the prices in a single equationthat is independent of any option pricing model. 18.-30. (canceled)